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The recently synthesized ThFeAsN iron pnictide superconductor exhibits a Tc of 30 K, the highest of the 1111-type series in the absence of chemical doping. To understand how pressure affects its electronic properties, we carried out microscopic investigations up to 3 GPa via magnetization, nuclear magnetic resonance, and muon-spin rotation experiments. The temperature dependence of the As75 Knight shift, the spin-lattice relaxation rates, and the magnetic penetration depth suggest a multiband s±-wave gap symmetry in the dirty limit, whereas the gap-to-Tc ratio Δ/kBTc hints at a strong-coupling scenario. Pressure modulates the geometrical parameters, thus reducing Tc as well as Tm, the temperature where magnetic-relaxation rates are maximized, both at the same rate of approximately -1.1K/GPa. This decrease in Tc with pressure is consistent with band-structure calculations, which relate it to the deformation of the Fe 3dz2 orbitals. © 2018 American Physical Society.